Exponential equations

Exponential functions and logarithms: Exponential functions

Exponential equations

There is an important rule we can use to solve exponential equations for an unknown variable $x$ .

$\blue{a}^\green{b}=\blue{a}^\purple{c}$

gives

$\green{b}=\purple{c}$

Example

$\begin{array}{rcl}\blue{3}^\green{x}&=&9\\\blue{3}^\green{x}&=&\blue{3}^\purple{2}\\ \green{x}&=&\purple{2}\end{array}$

Solve the equation for $x$ :
$3^{x+3}=81$

Do not use exponents and give your final answer in the form $x=\ldots$ .
Simplify the number at the dots as much as possible.

$x=1$

$\begin{array}{rcl} 3^{x+3}&=&81\\ &&\blue{\text{the original equation}}\\ 3^{x+3}&=&3^4\\ &&\blue{\text{write $81$ as a power of $3$}}\\ x+3&=&4\\ &&\blue{a^b=a^c\text{ gives }b=c}\\ x&=&1\\ &&\blue{\text{constant terms moved to the right}}\\ \end{array}$

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